which defines the enveloppe function for the re-weighted objective. For any \(a \in S^{d_\theta } = \lbrace  a \in \mathbb {R}^{d_\theta }, \Vert a\Vert =1 \rbrace \) , the surface of the unit sphere, define \(\sqrt{m} a^\prime (G_m^{(b)}(\theta )-G_n(\theta ))\) , a scalar-value Gaussian process defined on \(S^{d_\theta } \times \Theta \)  a compact subset of \(\mathbb {R}^{d_\theta } \times \mathbb {R}^{d_\theta }\) . Let \(\sigma _n^2 = \sup _{(a,\theta )\in S^{d_\theta } \times \Theta } \mathbb {E}^\star (m|a^\prime (G_m^{(b)}(\theta )-G_n(\theta )|^2) = \sup _{(a,\theta )\in S^{d_\theta } \times \Theta } a^\prime V_n(\theta ) a\)  which is finite and converges in probability to \(\sup _{\theta \in \Theta } [\lambda _{\text{max}}(\mathbb {E}[\nabla q(z_i,\theta )\nabla q(z_i,\theta )^\prime ])]\) , also finite. We can now apply results for scalar valued Gaussian processes. In particular, using Theorem 5.8 in [1]}, see also Proposition 3.19 in [2]}, we have for any \(u \ge 0\) :
\(&\mathbb {P}^\star \left( \sup _{(a,\theta )\in S^{d_\theta } \times \Theta }\sqrt{m}|a^\prime (G_m^{(b)}(\theta )-G_n(\theta )| \ge M_n + u \right) \le \exp \left( - \frac{u^2}{2\sigma _n^2}\right),\) 
